If I have understood correctly, for example an eletron orbiting an atom
does not have a precise position, but instead there exists a probability
function around the atom where the electron "is" at any given moment.
The probability is highest at a certain distance from the nucleus, but
extends well beyond that (even though obviously sharply reducing in
probability the farther from the atom we get).

  Also, if I have understood correctly, there's no bound to this probability
function. In other words, in principle the electron could be anywhere in
the entire universe at any given moment (although the odds get staggeringly
small when we get even slightly away from the atom.)

  So this got me thinking: That means that the probability function itself
cannot be quantized (because it would drop to zero after getting smaller
than the unit quant of "probability"). In other words, the probability
function itself is contiguous, not quantized.

  If this is so, it goes against the notion that basically *everything* in
this universe is quantized.

-- 
                                                          - Warp

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Warp wrote:
>   If this is so, it goes against the notion that basically *everything* in
> this universe is quantized.

Actually, what happens is this:

You calculate the probabilities, summing in smaller and smaller and smaller 
probabilities of something happening. Now if you do the math all the way 
down to zero probability, you find out the sum blows up to infinity. But as 
long as you stop somewhere *before* you go all the way down to zero, you get 
an answer proportional to where you stopped.  Proving that it didn't matter 
where you stopped as long as you didn't go all the way down to zero was what 
Feynman won the Nobel prize for. It's called "renormalization."

Basically, if you add the sequence of terms together, it (more or less, 
depending on the particles involved) rapidly converges on the answer. If you 
take the limit, tho, it diverges.  Weird.

So, mathematically, yeah, the probability itself seems like it's quantized, 
at least as far as I understand it.  The math that matches to 15 decimal 
places behaves as if there's a lower bound of "epsilon", but that isn't zero.

It has been too long since I read about it to remember the details, tho, so 
this might all be incorrect.

-- 
Darren New, San Diego CA, USA (PST)
   Serving Suggestion:
     "Don't serve this any more. It's awful."

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Darren New wrote:
> summing in smaller and smaller and 
> smaller probabilities of something happening. 

BTW. By this, I mean that you say "How can the electron get from A to B?"

Well, it could go in a straight line. That's the main factor.

Or it could absorb some photon from somewhere else on the way at point C in 
between, change direction, and end up at B.

Or it could emit a photon at point C, and end up at B.

Or it could absorb a photon at C, emit one at D, and end up at B.

Etc etc etc.

Each emit/absorb interaction happens with about the probability of 1/137, 
which is the "charge" of the electron. So you don't have to go too far to 
get a good number of decimal places. That last step adds about one part 
error per 18,000, so one decimal place out of 5 or so. (Quarks have a much 
higher "charge", like 1/8 IIRC, so you have to run the calculation out lots 
of steps before you get similarly good results arithmetically.)

But if you add up *all* those probabilities, for *every* possible C or D or 
etc, everything goes to infinity. If you stop after any number of 
interactions (i.e., assume space is discrete), the math says you get a 
really close answer. But if you add up *all* theoretical interactions 
(taking the limit as the distance between C and D goes to zero), the answers 
make no sense.

Now I know what you're saying. Even if space was quantitized to one inch, 
assuming an open universe, there should be places sufficiently far away from 
your starting point that the probability is arbitrarily close to zero.

However, it all works out OK if space is both quantitized and finite; then 
you get quantitized probabilities as well.

Or maybe the probability actually does go to zero far enough from the 
electron. I'm not sure what relativity says about electrons moving faster 
than light over significant distances. It may be that sufficiently far from 
the atom, the probability of the electron being there is absolutely zero, 
just like the probability of two electrons in the same quantum state being 
in the same place is absolutely zero.

I'll have to ask a friend I know who is a practicing theoretical physicist.

-- 
Darren New, San Diego CA, USA (PST)
   Serving Suggestion:
     "Don't serve this any more. It's awful."

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Warp <war### [at] tagpovrayorg> wrote:
>   If this is so, it goes against the notion that basically *everything* in
> this universe is quantized.

I know, we should vote for it.

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On 11/01/2011 01:22 AM, nemesis wrote:

> I know, we should vote for it.

As I understand it, Einstein's theory of relativity is basically 
"Suppose that nothing can exceed the speed of light. What would that imply?"

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Darren New <dne### [at] sanrrcom> wrote:
> Warp wrote:
> >   If this is so, it goes against the notion that basically *everything* in
> > this universe is quantized.

> Actually, what happens is this:

> You calculate the probabilities, summing in smaller and smaller and smaller 
> probabilities of something happening. Now if you do the math all the way 
> down to zero probability, you find out the sum blows up to infinity. But as 
> long as you stop somewhere *before* you go all the way down to zero, you get 
> an answer proportional to where you stopped.  Proving that it didn't matter 
> where you stopped as long as you didn't go all the way down to zero was what 
> Feynman won the Nobel prize for. It's called "renormalization."

  But if there is no lower limit (greater than zero) then it's not
quantized. A quantized system would have an exact minimum amount, and all
other amounts would be exact multiples of that minimum.

  If you can go arbitrarily close to zero, there's no quantization.

-- 
                                                          - Warp

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>    If I have understood correctly, for example an eletron orbiting an atom
> does not have a precise position, but instead there exists a probability
> function around the atom where the electron "is" at any given moment.
> The probability is highest at a certain distance from the nucleus, but
> extends well beyond that (even though obviously sharply reducing in
> probability the farther from the atom we get).
>
>    Also, if I have understood correctly, there's no bound to this probability
> function. In other words, in principle the electron could be anywhere in
> the entire universe at any given moment

Assuming the universe doesn't have infinite volume, that would mean 
there are only a finite number of positions in space the electron could 
be in - something like the volume of the universe divided by the planck 
volume.  If you know the shape of the probability function, you could 
probably figure out the minimum amount of non-zero probability.  But 
that wouldn't necessarily mean it was quantized though.

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Warp wrote:
>   But if there is no lower limit (greater than zero) then it's not
> quantized. A quantized system would have an exact minimum amount, and all
> other amounts would be exact multiples of that minimum.

Not necessarily. A photon can have arbitrarily close to zero energy, but 
it's still a quantum photon. You still never get half a photon.

In any case, as I said, I don't understand the math at that level, except to 
know there's something funky going on.

-- 
Darren New, San Diego CA, USA (PST)
   Serving Suggestion:
     "Don't serve this any more. It's awful."

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scott <sco### [at] scottcom> wrote:
> Assuming the universe doesn't have infinite volume, that would mean 
> there are only a finite number of positions in space the electron could 
> be in - something like the volume of the universe divided by the planck 
> volume.  If you know the shape of the probability function, you could 
> probably figure out the minimum amount of non-zero probability.

  Except that the universe is expanding, which means that the amount of
positions is increasing.

-- 
                                                          - Warp

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Warp wrote:
>   Except that the universe is expanding, which means that the amount of
> positions is increasing.

Again, I think this is another place where GR and QM are mathematically 
incompatible. It's hard to say what edge-case GR means to QM when we don't 
have an actual theory that relates the too.

But it's an interesting observation, yes.

-- 
Darren New, San Diego CA, USA (PST)
   Serving Suggestion:
     "Don't serve this any more. It's awful."

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